New Proth Search Page

Proth Search Page

Originally created by Ray Ballinger (1953‐2014)
Rebuilt and maintained by Wilfrid Keller

Contents
Primes k · 2ⁿ + 1	List of primes for k < 300 List of primes for 300 < k < 600 List of primes for 600 < k < 900 List of primes for 900 < k < 1200 Frequencies (extended and corrected) History
Fermat numbers	Standard factoring status Factors of generalized Fermat numbers GFN03 factoring status GFN05 factoring status GFN06 factoring status GFN07 factoring status GFN10 factoring status GFN11 factoring status GFN12 factoring status GFN double check (new) Search limits and reservations
Primes k · 2ⁿ − 1	List of primes for k < 300
The Sierpiński problem	Definition and status
The Riesel problem	Definition and status
Cullen primes	Definition and status Generalized Cullen primes (expanded)
Woodall primes	Definition and status

Last modified October 9, 2024.
Lists of primes k · 2ⁿ + 1 and k · 2ⁿ − 1 updated:
all currently known primes included.
GFNfacs.html now contains 9106 "new" divisibilities,
220 of which were found during this year.
2023 had again been a productive year for GFN factors:
470 divisibilities found!
Number of divisibilities discovered in previous years:
362 (2022), 747 (2021), 624 (2020), 41 (2019), 27 (2018),
18 (2017), 46 (2016), 79 (2015), 136 (2014), 213 (2013),
457 (2012).

July 4, 2024:
New factor of Fermat number F(223380) !

May 24, 2024:
New factor of Fermat number F(251) !

November 27, 2023:
Frequencies table corrected for 1200 < k < 10000:
adjusted count for 80000 < n ≤ 90000.

Four new factors of Fermat numbers found within one
month of 2023: F(31880) (July 10), F(4192909) (July 17),
F(160) (July 27), F(699760) (July 28).

June 25, 2023:
New factor of Fermat number F(425).

May 31, 2023:
New factor of Fermat number F(1784).

April 23, 2023:
Candidate of Riesel Problem eliminated.

April 21, 2023:
New factor of Fermat number F(1436).

April 16, 2023:
Frequencies table corrected for 1200 < k < 10000: adjusted
counts for 100000 < n ≤ 200000 and 400000 < n ≤ 500000.

February 9, 2023:
New factor of Fermat number F(1493).

December 2022:
Factorization of numbers GF(m,a) and xGF(m,a,b),
b < a ≤ 12, completed for all m ≤ 8.

November 27, 2022:
Candidate of Riesel Problem eliminated.

July 25, 2022:
New largest known factor of a Generalized Fermat
number found: 7 · 2^20267500 + 1 divides GF(20267499,12).

March 8, 2022:
A second long-term omission was detected in the list
of primes k · 2ⁿ + 1 : the prime 281 · 2^2051865 + 1 had
to be added.

November 25, 2021:
Candidate of Extended Sierpinski Problem eliminated.

November 24, 2021:
New factor of Fermat number F(1379).

August 11, 2021:
New factor of Fermat number F(66643).

June 29, 2021:
First factor of GF(9,11) found, a P52 prime.

May 2021:
A long-term omission was detected in the list of primes
k · 2ⁿ + 1 : the prime 879 · 2¹¹⁰⁰⁷⁵ + 1 had to be added.

Five Riesel problem candidates eliminated within less
than six months:
k = 146561 (16 Nov 2020), k = 9221 (07 Feb 2021),
k = 2293 (13 Feb 2021), k = 192971 (07 Mar 2021),
k = 206039 (26 Apr 2021).

March 27, 2021:
6896-digit final factor of xGF(13,7,5) proven prime.

March 12, 2021:
First factor of GF(14,10) found, a P37 prime.

March 1st, 2021:
New factor of Fermat number F(40).

December 31, 2020:
In 2020, 612 GFN divisibilities were found:
501 by Gary Gostin, 111 by others.

October 05, 2020:
New largest known factor of a Fermat number
found: 7 · 2^18233956 + 1 divides F(18233954).
The same prime also divides GF(18233952,7), the largest
known composite GF(m,a) number for any basis a ≤ 12.

July 06, 2020:
Summary of frequencies corrected and extended.

April 01, 2020:
Largest known divisor of a Generalized Fermat number
found: 9 · 2^13334487 + 1 divides GF(13334485,3),
GF(13334486,7) and others.

February 11, 2020:
Gary Gostin starts generating new GFN divisibilities:
82 factors delivered at once.

February 10, 2020:
New factorization at GFNsmall.html:
Cofactor of 10^(2^8) + 3^(2^8) is P85 × P105.

January 22, 2020:
New factor of Fermat number F(5523858):
is new largest known factor of a Fermat number.

December 25, 2019:
Candidate of Extended Sierpinski Problem eliminated.

December 9, 2019:
New factor of Fermat number F(2587), which brings to 350
the total number of known factors.

October 16, 2019:
New factors of Generalized Fermat numbers GF(4800313,3)
and GF(4800310,5).

October 14, 2019:
New factor of Generalized Fermat number GF(4673541,7).

October 13, 2019:
New factor of Generalized Fermat number GF(4532462,11).

September 29, 2019:
New factor of Generalized Fermat number GF(3964696,11).

July 31, 2019:
Generalized Fermat number GF(25,10) proved composite.

July 6, 2019:
Generalized Fermat number GF(24,10) proved composite.

March 23, 2019:
New factor of Fermat number F(9863).

January 28, 2019:
New factor of Fermat number F(118).

December 31, 2018:
Table of factors GFNfacs.html now includes 6608
new divisibilities, 21 of them found in 2018.

December 19, 2018:
New factor of Fermat number F(132).

December 13, 2018:
New factor of Fermat number F(3345).

November 2, 2018:
New factor of Fermat number F(2144).

July 24, 2018:
New factor of Fermat number F(5199), which brings to 300
the total number of known composite Fermat numbers.

May 1, 2018:
New factor of Generalized Fermat number GF(10746,12).

April 5, 2018:
New factor of Fermat number F(274).

April 3, 2018:
Candidate of Extended Sierpinski Problem eliminated.

March 10, 2018:
New factor of Fermat number F(63480).

January 15, 2018:
New factorization at GFNsmall.html:
Cofactor of 11^(2^8) + 10^(2^8) is P62 × P107 .